The 24-year-old came from one set behind to overcome the five-time champion, as she inflicted Williams's earliest loss in the competition since 2005.
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Williams lost serve in the opening game of the match, but soon recovered following a lengthy rain delay to comfortably take the first set.
Cornet raced into a five-game lead in the second, before eventually prevailing 6-3 to ensure the contest was level heading into the third and final set.
World number 24 Cornet grabbed two breaks of serve in the decider to come out of top, ending her American opponents hopes of claiming the title once more.